1 The Limits of Knowledge: Philosophical and practical aspects of building a quantum computer Simons Center November 19, 2010 Michael H. Freedman, Station Q, Microsoft Research 2 • To explain quantum computing, I will offer some parallels between philosophical concepts, specifically from Catholicism on the one hand, and fundamental issues in math and physics on the other. 3 Why in the world would I do that? • I gave the first draft of this talk at the Physics Department of Notre Dame. • There was a strange disconnect. 4 The audience was completely secular. • They couldn’t figure out why some guy named Freedman was talking to them about Catholicism. • The comedic potential was palpable. 5 I Want To Try Again With a rethought talk and broader audience 6 • Mathematics and Physics have been in their modern rebirth for 600 years, and in a sense both sprang from the Church (e.g. Roger Bacon) • So let’s compare these two long term enterprises: - Methods - Scope of Ideas 7 Common Points: Catholicism and Math/Physics • Care about difficult ideas • Agonize over systems and foundations • Think on long time scales • Safeguard, revisit, recycle fruitful ideas and methods Some of my favorites Aquinas Dante Lully Descartes Dali Tolkien • Lully may have been the first person to try to build a computer. • He sought an automated way to distinguish truth from falsehood, doctrine from heresy 8 • Philosophy and religion deal with large questions. • In Math/Physics we also have great overarching questions which might be considered the social equals of: Omniscience, Free Will, Original Sin, and Redemption. 9 10 • P/NP ↔ Omniscience • Quantum Mechanics ↔ Redemption • Universality ↔ Original Sin • Unicity ↔ Free Will 11 (1) P/NP (Omniscience) • The limits of knowledge • The limits of computation • The scaling of effort required to: ▫ solve problems (factor numbers) ▫ discover proofs Kurt Gödel 12 Quantum Mechanics (Redemption) • Does it provide a complete framework for all physics? Does it redeem our understanding and give us a theanthropic perspective of the world? • What about gravity? • What about the classical world? ▫ the measurement problem ▫ where do unrealized probabilities go? Schrodinger 13 Universality (Original Sin) Ken Wilson • Physical systems (when viewed from a distance) can be grouped into a small number of classes with identical scaling laws. • “Curse of Universality.” By looking at the emergent structure, one may never know what microscopics caused it. 14 Unicity (Free Will) Descartes “The best of all possible worlds.” – Gottfried Leibniz • Did our universe have to be roughly as it is? ▫ 3+1 large dimensions ▫ stable matter ▫ weakly chaotic dynamics • Did our universe have to be exactly as it is? Is it preordained? 15 The topic today is: Math Physics QC Computer Science 16 And also: Math Physics QC Computer Science Everything Else 17 • Along the way we’ll encounter our dancers: Unicity P/NP Quantum Mechanics Universality Gödel Ken Wilson Scott Aaronson 18 • What we can hope to compute is limited by the scaling of effort inherent in each type of problem. • Obviously it is a lot more work to factor a large number than a small one. • But exactly how fast does the work load grow? 19 NP P PSPACE Q factoring The class Q depends on a different way, a quantum mechanical way, of storing numbers. 20 Quantum computation is a new paradigm in which computational work obeys different scaling laws than those that are known to hold in present day “classical” computers. 21 Modern Church-Turing (MCT) Thesis: • There are only two physically realistic models of computation: ▫ One based on Classical Physics ▫ One based on Quantum Physics Alonzo Church Alan Turing 22 Corollary of MCT • All we will ever know (or at least compute) will lie in Q. 23 Unicity (Free Will): Could the universe have been different? • Is there a world where NP-complete problems can be solved efficiently? • Many (Aaronson) think not – that just like perpetual motion, such worlds cannot be consistent. Free Will! Indeed. 24 So perhaps there is only one possible theory governing our universe: Quantum Mechanics 25 Q: What new power is conveyed by computing quantum mechanically? A: Superposition 26 • Superposition means that a general state ψ may be written as a linear combination of eigen-states ψi, which typically are classical configurations ψ = ∑ αi ψi • The coefficients αi, called amplitudes, are “square roots” of probabilities: 2 ∑ |αi| = 1 27 • Square roots of probabilities are not intuitive. • Nothing in our large scale clumsy world, nothing in our evolutionary experience, prepares our mind for superposition. • Superposition was born amid mystery and seeming paradox in the period 1900-1927. Plank Born Bohr Heisenberg Schrodinger Radiation, Diffraction, Scattering, Atomic Spectra 28 • The double slit experiment shows amplitudes at work. 29 + All closed - 1 open + 2 open Blind Screen -|α|2 0 +|α + β|2 -|β|2 Source |α + β|2 = |α|2 + |β|2 Observed pattern “Classical expectation” 30 How do amplitudes, opposed to probabilities, enhance computational power? Peter Shor In a cleverly designed algorithm factoring useless computational paths can often be arranged to cancel out – like the dark spots (“nodes”) in the double slit experiment – and not consume computational resources. This is possible because amplitudes, unlike probabilities, can be negative (or even imaginary). With quantum effects, “factoring” goes from exponential to polynomial time. 31 What did Shor do? • Classical part: Order finding: factoring ▫ Suppose f(x) = ax mod N has period r (even). ▫ Then (usually): (ar/2 + 1)(ar/2 - 1) = ar – 1 = kN will “separate” factors of N. 32 • Quantum Part: Study f(x) = ax mod N ^ 2휋kxi/M ▫ In Fourier space: f(k) = Mk,x f(x), Mk,x = e , M is approximately N2 ^ ▫ Observing f returns information on the periodicity of f ▫ The nodes or “dark spots” are the places where f^( k) is small. 33 What might a quantum computer do in the real world? • (1) Wreak havoc: Break all classical codes Panic on Wall Street 34 • (2) Allow physicists to explore exotic states of matter ▫ Strongly correlated electron systems High Tc superconductors 2-dimensional electron gasses (2-DEGS) Exotic magnets • ? Compute string theories ? (good research problem!) 35 • (3) Sample from the solution space of exponentially large linear systems. Many engineering applications: Electrical engineering and communication Optimization Fluid flow? 36 • (4) Allow chemists / pharmacologists to design drugs? 37 • (5) Artificial intelligence? ▫ In 1950, Alan Turing predicted : Computing power would grow fast (it grew faster) By 2000 we would have a hard time saying that machines were not thinking. (did not happen) Quantum computing may give AI a second chance 38 Topology • There is a topological approach to quantum computation that avoids local degrees of freedom: nuclear spin, electron spin, photon polarization, etc. • We need a two dimensional electron gas (2DEG)– with a special property • Majorana fermions localized in “vortices” 39 There are two prime candidates for Majoranas: • Fractional Quantum Hall Effect (FQHE) at ν = 5/2 • px + ipy superconductors 40 • We can execute operations (“gates”) on system state by braiding Majoranas. • Or “quasi particle interferometry.” 41 42 43 Fractional Quantum Hall Effect • 2DEG • large B field (~ 10T) • low temp (< 1K) • gapped (incompressible) • quantized filling fractions nh, RR 1 , 0 m xy e2 xx 44 A topological state of matter: the quantum Hall state e2 n xy h • Topological origin of the quantized Hall conductance: • Bulk gap (Landau level gap) • The first Chern number (Laughlin PRB 1981, Thouless, Kohmoto, Nightingale, den Nijs (TKNN), PRL 1982) E • Chiral edge states on the boundary QH x 45 5/2 Hall Fluid 46 • FQHE physics is topological, meaning that distance plays no role (or at least an inferior role). • Topology is “rubber sheet geometry” and FQHE is “rubber sheet physics.” • Controlled by the Chern-Simons lagranian which does not mention distance! • It is topologically invariant 47 Topological-invariance is clearly not a symmetry of the underlying Hamiltonian. How can Chern-Simons theory possibly describe the low energy physics of the above Hamiltonian? 48 The answer goes back to: 1970: Wilson, Renormalization: How does the Langrangian evolve when re-expressed using longer length scales, lower frequencies, colder temperatures? The terms with the fewest derivatives dominate: This is because in momentum space, differentiation becomes multiplication by k and: 2 k>>k 49 • Chern-Simons Action: A d A + (A A A) has one derivative, • while kinetic energy p2/2m is written with two derivatives. (pi = 1/2 m d/dxi ) • Thus, in condensed matter at low enough temperatures, we may expect to see systems in which topological effects dominate and geometric detail becomes irrelevant. • FQHE is such a system. 50 A new proposal: Majoranas in a px + ipy superconductor within a “conventional” semi-conductor device 51 52 53 54 55 56 • In the topological approach, interferometers will play a key role: 57 FQH interferometer Willett et al. `08 for =5/2 58 A lot of theory [Bonderson, et al] has been devoted to using interferometers to: • Measure topological charge • Manipulate quantum information • Simulate quasi particle braiding FQH fluid (blue) 59 60 • Interferometry creates probabilistic combinations i of quasi-particle world lines. e •Let me take you through some cartoons: = ei ' 61 • Braiding, and hence calculation, can be simulated by measurement. 1 1 a (23) a 1 (34) 1 (13) 1 (23) 1 a a 62 Measurement Simulated Braiding! a a (23) (34) (13) (23) (14) 1 1 1 1 R a a 63 • Bob Willett of Bell Labs has presented evidence of Majoranas in ν=5/2 FQHE systems Robert Willett 64 Bob Willett 24 hrs/run 65 Conclusions: • In building a topological quantum computer, universality is our friend. ▫ It allows us to model and study exotic states of matter such as the fractional quantum Hall effect.